Which End Do the Bullets Go in Again

You know I similar the MythBusters, right? Well, I take been meaning to wait at the shooting bullets in the air myth for quite some time. At present is that fourth dimension. If you didn't catch that particular episode, the MythBusters wanted to run across how unsafe it was to shoot a bullet directly upwards in the air.

I am non going to shoot whatsoever guns, or even driblet bullets - that is for the MythBusters. What I will exercise instead is make a numerical calculation of the motion of a bullet shot into the air. Here is what Adam said about the bullets:

• A .30-06 cartridge will become 10,000 anxiety high and have 58 seconds to come back down
• A 9 mm will go 4000 feet and take 37 seconds to come back downwards.

Adam was besides able to experimentally make up one's mind that both the 9 mm and the .thirty-06 have a terminal speed of almost 100 mph. Then, that is what I have to work with. Oh - also, they measured how far a 9mm bullet penetrated into the dirt (but they couldn't find the .30-06 ones).

The programme

This is actually like to Hancock throwing a boy. The bones plan is to use a numerical adding to model the motility of a bullet. Afterwards the bullet leaves the gun, it has forces acting on it like this:

I made 2 force diagrams because the air resistance strength is going to be in opposite direction as the motion. This means that moving up bullet volition expect different than going downwards. So, this trouble seems simple enough - right? I have actually done this earlier (here is an example of the air resistance on a football). But in this case, in that location are some other things to consider.

• Does the normal model of air resistance work (being proportional to five²)?
• What is the drag coefficient of a bullet?
• What nearly the density of air? Do I need to take that into business relationship?
• What most the alter in gravitational field of the Globe equally the bullet moves upward?

Numerical Modeling

I don't desire to get into the details, but in case you forgot, the numerical calculation works this mode:

• Break the motion into tiny little time steps. During these steps, I can pretend (assume) that the force is constant. With a small-scale enough time, this is true enough.
• For each time step: Summate force
• Calculate modify in momentum (assuming constant force)
• Calculate modify in position (assuming constant momentum)
• repeat

If you want more details on numerical calculations, bank check out this basic post.

Starting info

I am just going to look at the .30-06, only I need some ballistics info. Here is what I found (wikipedia, of grade)

• Slug mass = 9.7 grams
• Muzzle velocity = 880 m/s (actually, this is just the fastest - the slowest is 760 m/south and 14 thousand - not sure which the Mythbusters used)
• Final velocity = 44.7 g/s

Air Resistance

If I desire to model the air resistance, I can use the following:

The problem is that bullets go really fast. I mean really fast. It is non safe to assume that the drag coefficient (C) is abiding with speed. Wikipedia comes to the rescue once again. In this case, in that location is this very useful table:

Apparently, there is lots of argue near the air drag of a bullet. I will but utilise the tabular array above to make variable drag coefficient. So, that is C, I tin find the constructive area by looking at concluding velocity. At terminal velocity, the weight = air resistance and then:

Using the known values for mass, yard, C (from the tabular array) and the density of air (at sea level), I get an surface area of A = iii.45 ten 10^-4 m². Wikipedia lists the bullet as having a bore of 7.823 mm - this would give an area of 1.nine x 10^-4 m². I estimate these are kind of in the same brawl park. Well, there is a mode to test which is correct - but I volition showtime with the one from the terminal velocity.

Density of Air

This is starting to become complicated. Good thing I am making a reckoner practice all the work. If the MythBusters are right and the bullet goes x,000 feet loftier, then I volition need to look at the change in the density of air. Here is an explanation of the density with altitude calculation. Using this expression (which I am not showing because it is slow), I tin plot density equally a function of distance. This is it:

Gravity dependence on summit

Of course the gravitational field is not abiding with superlative, but is it close enough? The real gravitational field (one thousand) is:

Where Thousand is the universal gravitational constant, thou_E is the mass of the Earth, R_E is the radius of the Globe, and h is the height above the surface. What would the value of g be at 4000 meters? (the MythBusters said the bullet went ten,000 feet - about 3000 meters). Or rather, what would be the percent difference betwixt the surface and 3000 meters upwardly? It is 99.9% the value at the surface. I can simply pretend its constant.

Now for the adding:

Here is a plot of the vertical position of the bullet every bit a part of fourth dimension, shot directly upwardly.

Well, that doesn't agree with the MythBusters' model. What if I go with the smaller area value?

Meliorate, simply still does not agree? I could endeavor a different bullet. Permit me effort the 1 with the lower muzzle velocity, but higher mass. I volition use a mass of xiv grams and an initial velocity of 760 m/southward. This gives a max summit of about 1300 meters with a total fourth dimension of about 34 seconds.

I remember I come across some other mistake. My table of drag coefficients are matched upward with mach number, not velocity. If I increase my altitude, that changes the speed of sound - doh! Ok, I don't recollect this matters too much. Here is a speed of audio calculator. It'southward from NASA, so it has to exist skillful, right? Anyway, information technology says the speed of sound at body of water level is 340 m/due south, at 5000 meters information technology is 320 m/s. Instead of calculating the speed at every pinnacle, I merely changed the speed of sound to 320 grand/s. It doesn't actually alter the max height.

Perchance the problem is with the drag coefficient. Hither is a plot of the elevate coefficient (C) as a part of speed.

It looks "blocky" because I am just using data from that wikipedia table. But maybe this is the problem. Actually, possibly the problem is that the elevate coefficient table doesn't work very well at low (very depression) speeds.

Mayhap this isn't even incorrect

Now that I retrieve nearly it, the MythBuster'southward said they false the .xxx-06, but when they shot it in the air, they never heard nor institute the bullets. Who knows how long it took. They did know the time for the 9mm bullets, they heard them striking the ground. Permit me run my calculations with the 9mm info. Using mass of 7.45 grams and initial velocity of 435 m/southward, I get:

Which seems much closer to what they (MythBusters) had. And I just realized another fault on the .30-06. I calculated the surface area with the diameter instead of the radius.

See. That is amend. I hope this is a lesson to all you kids out there. Heed your factor of two's. Of class if I get this to work, at present my terminal velocity is much higher than what they measured. Oh well.

My next stride is to look at the final speed of the bullet if yous shoot it non straight up. I suspect this is the how people get killed.

tuckerwithand.blogspot.com

Source: https://www.wired.com/2009/09/how-high-does-a-bullet-go/

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